The invention of the gear precedes Aristotle or approximately 330 B.C. Early gears, intended for operation at low speed or low loads, such as in clockworks, were initially formed by hand with simple tools and jigs. The exact shape of the gear tooth and the effect of shape on strength and durability was of little importance. Early gears used in power transmission applications, in mills and the like, achieved greater strength from increased size and improvements in materials, rather than from a close attention to the shaping of the gear teeth.
Nevertheless, the mathematics of gear tooth shapes was worked out at an early date. These early mathematical analyses suggested that either of two different shapes would be desirable for gear teeth. Both shapes were in the family of mathematical curves called trochoids. The first shape was the epicycloid which is the curve traced by a point on the circumference of the circle as that circle rolls on the outside of a second circle without slipping. The second shape was the involute, which is the curve traced by a point on a line when the line is rolled without slipping on a circle.
Extended mathematical analysis of gear teeth begins effectively with Phillipe de La Hire, in 1694. In addition to his treatise, he designed gearing for a large water works. The treatise discussed the whole family of epicycloids and reached the conclusion that the involute curve was the best of all the exterior epicycloids. The gap between theory and practice is vividly shown by the fact that the involute form for gear teeth was not adopted in practice for 150 years. Mathematical analysis was further advanced by Euler (1754-55), but the involute curve was not mentioned by Camus (1733, 1766) in a treatise that had wide currency. Kaestner, writing in 1781, thought of his work as a systematic treatment of a stable and complete body of knowledge. This treatise may, therefore, be accepted as the close of the work on the abstract analysis of the geometry of motion in wheels and gears. (emphasis supplied) See, Abbott Payson Usher writing in The History of the Gear Cutting Machine. Robert S. Woodbury, p. ii, The Technology Press, MIT (1958) PA0 "The difficult mathematics of gears and gear cutting for any tooth on any axis has been almost completely solved. Some of the better known contributors to this work of modern gear theory are Ernest Wildhaber, Earle Buckingham, H. E. Merritt (England), Allen Candee, Werner Vogel, Hillel Poritsky, G. Niemann (Germany), Darle Dudley, Oliver Saari, Charles B. King and Meriwether Baxter". The Evolution of the Gear Art, Darle W. Dudley, p. 65, AGMA (1969).
With the advent of steam power in the early 1800's (and the concurrent need for gears suitable for higher speeds and loads) widespread interest in the exact shape of gear teeth was awakened among manufacturers. By approximately 1840, the involute had been adopted as superior to the epicycloid based on the work of John Hawkins and Robert Willis. Hawkins demonstrated, among other advantages, that the involute shape allowed more than one tooth to be engaged at a given time and further decreased the strain and the sliding to about one-half of that for a similar epicycloidal tooth. Thus, the involute provided more durable teeth.
Although both epicycloidal and involute gears were cut as late as 1880, in 1867 Brown and Sharp brought out a set of formed cutters for involute teeth and by 1898 a survey in American Machinist in Mar. 1898 indicated a near universal acceptance of the involute tooth. Woodbury, at 35.
Two notable exceptions to the near universal use of the involute for power transmission gears are the Wildhaber-Novikov helical gear and the Wildhaber "best circle" circular-arc gear. In both of these gears the gear tooth profile is a circular arc, not an involute. Generally, two meshing gears with teeth that have circular-arc profiles will not have conjugate contacting surfaces. Conjugate tooth profiles are those which allow the transmission of uniform rotary motion from a driving gear to a driven gear. More precisely, the condition of conjugacy requires that: (1) the ratio of the velocity of the two gears in mesh be constant and (2) that the ratio of the velocity of the two gears be equal to the ratio of the numbers of teeth of the two gears.
In the Wildhaber-Novikov helical gear, however, the three dimensional geometry of the helix serves to provide for the smooth transmission of power with a circular arc despite the lack of conjugacy in two dimensions. Hence, conjugacy is obtained when the surface is considered along the axis of the gear.
The Wildhaber "best circle" circular arc gear is not conjugate even in three dimensions. This gear is not suitable for constant use in high power applications but has a great advantage in being simple to manufacture and finds use in applications where wear and durability are of lesser importance. In the Wildhaber circular arc gear, the driven member will lag behind its correct position for either direction of rotation.
With these exceptions, however, the involute shape is generally established for power transmission gears and further investigations into tooth shapes were relatively limited. In 1969, it was noted:
A basic problem with gears based on the trochoids is that when the tracing point coincides with the rolling point the traced curve has infinite curvature, a condition that is unacceptable in practice. In involute gears, this means that the usable tooth height is limited to the space between the base circles; in order to get sufficient height for motion carryover it is necessary to avoid small tooth-numbers. Even then, the large profile curvature of the involute near the root of the tooth seriously reduces its load-carrying ability. Therefore, as will be seen, the involute is not optimal for applications where durability is critical.